We study semigroup actions on a coarse space and the induced actions on the Higson corona from a dynamical point of view. Our main theorem states that if an action of an abelian semigroup on a proper coarse space satisfies certain conditions, the induced action has a fixed point in the Higson corona. As a corollary, we deduce a coarse version of Brouwer’s fixed-point theorem.

Let $\Omega \left(n\right)$ denote the number of prime divisors of $n$ counting multiplicity. One can show that for any positive integer $m$ and all $j=0,1,\dots ,m-1$, we have

$$\#\{n\le x:\Omega (n)\equiv j(modm\left)\right\}=\frac{x}{m}+o\left(x^{\alpha }\right),$$

with $\alpha =1$. Building on work of Kubota and Yoshida, we show that for $m>2$ and any $j=0,1,\dots ,m-1$, the error term is not $o\left(x^{\alpha }\right)$ for any $\alpha <1$.

The dilogarithm identities for the central charges of conformal field theories of simply laced type were conjectured by Bazhanov, Kirillov, and Reshetikhin. Their functional generalizations were conjectured by Gliozzi and Tateo. They have been partly proved by various authors. We prove these identities in full generality for any pair of Dynkin diagrams of simply laced type based on the cluster algebra formulation of the Y-systems.

For smooth linear group schemes over $\mathbb{Z}$, we give a cohomological interpretation of the local Tamagawa measures as cohomological periods. This is in the spirit of the Tamagawa measures for motives defined by Bloch and Kato. We show that in the case of tori, the cohomological and the motivic Tamagawa measures coincide, which proves again the Bloch-Kato conjecture for motives associated to tori.

We show that every Kato surface admits a locally conformally Kähler metric.

We consider the problem of whether the norm one torus defined by a finite separable field extension $K/k$ is stably (or retract) rational over $k$. This has already been solved for the case where $K/k$ is a Galois extension. In this paper, we solve the problem for the case where $K/k$ is a non-Galois extension such that the Galois group of the Galois closure of $K/k$ is nilpotent or metacyclic.

It is shown that if $\phi$ denotes a harmonic morphism of type Bl between suitable Brelot harmonic spaces $X$ and $Y$, then a function $f$, defined on an open set $V\subset Y$, is superharmonic if and only if $f\circ \phi$ is superharmonic on ${\phi }^{-1}\left(V\right)\subset X$. The “only if” part is due to Constantinescu and Cornea, with $\phi$ denoting any harmonic morphism between two Brelot spaces. A similar result is obtained for finely superharmonic functions defined on finely open sets. These results apply, for example, to the case where $\phi$ is the projection from ${\mathbb{R}}^N$ to ${\mathbb{R}}^n$ ($N>n\ge 1$) or where $\phi$ is the radial projection from ${\mathbb{R}}^N\setminus \left\{0\right\}$ to the unit sphere in ${\mathbb{R}}^N$ ($N\ge 2$).

The aim of this paper is to describe the geometry of the generic Kummer surface associated to a $(1,2)$-polarized abelian surface. We show that it is the double cover of a weak del Pezzo surface and that it inherits from the del Pezzo surface an interesting elliptic fibration with twelve singular fibers of type $I_2$.